# Constructing wavefunction for a mixed state

This question is somehow the reverse of another question.

If a quantum system $$S$$ is in a pure state, then we can find a wavefunction that describes $$S$$. This wavefunction is unique up to a phase factor. We can find it from the density matrix.

This is not possible when $$S$$ is in mixed state. However, I expect that for every mixed state, $$S$$ is entangled to some system $$T$$ and the composite system $$S+T$$ can be described by a wavefunction.

For example: suppose that we prepare a spin system $$S$$ to be in the mixed state with probability 0.5 for spin-up and probability 0.5 for spin-down. Then we can not describe $$S$$ by a single wavefunction. But during the process of preparing $$S$$, we used some physical process $$T$$ to generate the probabilities of 0.5. Say that $$T=|0\rangle$$ would generate spin-up and $$T=|1\rangle$$ would generate spin-down. Then the composite system $$S+T$$ has the wavefunction $$\psi = \frac12\sqrt2 |\uparrow\rangle \otimes |0\rangle+\frac12\sqrt2 |\downarrow\rangle \otimes |1\rangle$$.

Is it possible to find such a $$T$$ for all mixed states $$S$$?

• Yes. It is called purification. See e.g. Wikipedia. Commented Jan 22 at 13:00
• Thank you, that was the idea I was looking for! Commented Jan 22 at 13:18
• Consider to answer your question yourself, this might help potential future readers with a similar question/problem. Commented Jan 22 at 13:23